Problem: The equation of hyperbola $H$ is $\dfrac {(y+9)^{2}}{16}-\dfrac {(x-3)^{2}}{25} = 1$. What are the asymptotes?
We want to rewrite the equation in terms of $y$ , so start off by moving the $y$ terms to one side: $\dfrac {(y+9)^{2}}{16} = 1 + \dfrac {(x-3)^{2}}{25}$ Multiply both sides of the equation by $16$ $(y+9)^{2} = { 16 + \dfrac{ (x-3)^{2} \cdot 16 }{25}}$ Take the square root of both sides. $\sqrt{(y+9)^{2}} = \pm \sqrt { 16 + \dfrac{ (x-3)^{2} \cdot 16 }{25}}$ $ y + 9 = \pm \sqrt { 16 + \dfrac{ (x-3)^{2} \cdot 16 }{25}}$ As $x$ approaches positive or negative infinity, the constant term in the square root matters less and less, so we can just ignore it. $y + 9 \approx \pm \sqrt {\dfrac{ (x-3)^{2} \cdot 16 }{25}}$ $y + 9 \approx \pm \left(\dfrac{4 \cdot (x - 3)}{5}\right)$ Subtract $9$ from both sides and rewrite as an equality in terms of $y$ to get the equation of the asymptotes: $y = \pm \dfrac{4}{5}(x - 3) -9$